Ampere's law
1. Ampere's law: Circular path
We have seen that the magnetic field produced by
a long straight wire crossed by a current I at a point P
in the axis of the loop is:
B = (μoI/2πR)
R is the perpendicular distance from the wire to the point P. The
direction of the magnetic field B is tangent to a line of magnetic
field. Each line closes on itself and encircles the current carrying
wire. The related circle is in a plane perpendicular to the wire.
This circle is considered as a closed path, and its circumference
2πR is equal to ∫dr; where dr is the distant element along
the circumference of radius R. Hence the above relation becomes:
B 2πR = B ∫dr = ∫Bdr μoI
Using a dot product, we generalize this result to a
line integral of B formula called Ampere's law:
∫ B.dr μoI
dr is the infinitesimal displacement along the closed path. The
line integral does not depend on the radius of the closed
circle.
2. Ampere's law: Arbitrary closed path
Consider another closed path linked by the current in
a long straight wire. The closed path lies in the plane perpendicular
to the axis of the wire. It consists of several circular arcs centered
on the axis of the wire and several radial lines that connect
the arcs.
Since the line integral of B does not depend on the distance
(radius) of the circular arcs, and the dot product of B and
the radial lines is zero, the contributions to the
magnetic field around the wire is the sun (integral) along the
circular arcs ln(arc length subtended the angle
θn = ln/Rn):
∮Bdr =
∮(1)B1dr + ∮(2)B2dr +
... ∮(n)Bn dr =
B1∮(1)dr + B2∮(2)dr +
... Bn∮(n)dr =
l1B1 + l2B2 +
... + lnBn =
ΣlnBn
We have:
circular arc n = ln = Rnθn, and
Bn = μoI/2πRn
Hence
lnBn = Rnθn μoI/2πRn =
θn μoI/2π
Σ lnBn = μoI/2π Σ θn =
μoI/2π (2π) = μoI
Therefore:
3. Ampere's law: Path not linked by current
Now, we consider a closed path that is not linked by
the current in the long straight wire. In the case (a) of the
closed path with two circular arcs that subtend the same
angle θ at their common center, and two radial lines,
the contribution of the radial lines is again zero, and the
only contribution comes from the circular arcs part:
∮B.dr = ∮B.dr + 0 +
∮B.dr + 0 =
B1 l1 - B2 l2 = μoI/2πRn
We have:
l1 = R1θ,
l2 = R2θ
and
B1 = μoI/2πR1
B2 = μoI/2πR2
B1 l1 = μoI l1 /2πR1 =
μoI l1 l2 /2πR2l1 =
μoI l2 /2πR2 = B2 l2
Therefore
∮B.dr = 0
For the path that is not linked by the current: the line integral of B
around any closed path is zero:
→ →
∮ B . dr = 0
4. General expression of Ampere's law
Consider an arbitrary closed path near currents of conductors of any form.
The arbitrary closed path is linked by some currents (I1, I2, I3, I4) and
not linked by some others (I5, I6). The sum Σi of the currents that
link the closed path gives the general line integral of the magnetic field
B around this path:
5. Ampere's law and the displacement current
Let' rewrite Ampere's law:
→ →
∮B . dr = μoΣi
Note that the origin of this formula comes from the result of
considering a long straight wire. The question is what about the fact
that the wire is not long? Secondly, this law
involves curved path that is linked by the current piercing
a surface without specifying that surface.
The inconsistency
of this law was raised by James Clerk Maxwell in 1861 because
he needed something else to complete his equations for the
electromagnetic waves. He suggested adding another current
called displacement current Id to the conduction
current I in order to write a new law called Maxwell-Ampere
law . The expression of Id is:
Id = εodΦE/dt
ΦE is the flux of the electric field
E through the surface bounded by the closed path linked by the
displacement current Id.
The only given proof to bring out the existence of this current
is to consider a capacitor. If a capacitor is being charged, then
there is a change in the electric field E between the plates of the
capacitor due to the accumulation of charges Q on the plates of
area S = Q/σ where σ is the surface charge density.
Since E = σ/εo = Q/Sεo,
the displacement current is Id = dQ/dt = d(ESεo)/dt =
εo d(ES)/dt = εo dΦE/dt.
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∮ |
B ⋅ |
dr |
= |
μo (I + εodΦE/dt) |
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